How to highlight intersection of meshes?
Yes, booleans can be used here. It should be noted that booleans can be rather glitchy in many cases. There are ways to mitigate them, but the methods I know of won't apply very well to the type of animation you're looking for.

Create a large cube that encompasses all of your objects. Then, apply two Intersect Boolean modifiers, one for each of the intersecting objects.

The first boolean will cutout the cube into something that resembles one of your objects (I say resembles simply due to the glitchy nature of booleans), and the second intersect will take the cube (that now is one of your objects), and cuts out whatever is not touching the second.

Here's a quick demo of that: